On the Number of Equivalence Classes of Boolean and Invertible Boolean Functions

Miodrag Zivkovic,Marko Caric

doi:10.1109/tit.2020.3025767

Miodrag Zivkovic, Marko Caric

https://doi.org/10.1109/tit.2020.3025767

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The number U n of equivalence classes of Boolean functions of n variables and the number V n of equivalence classes of vectorial Boolean functions of n variables under the action of four groups of transformations are considered. The four groups are the group S n ' of permutations of variables, the group G n of permutations and complementations, the linear group GL(n, 2) and the affine group AGL(n, 2). Harrison obtained cycle indexes for these groups and the expressions for U n and V n in terms of the corresponding cycle index. He also tabulated the numbers U n , V n and the cycle indexes for n≤6 for S n ' and Gn, and for n≤5 for GL(n, 2) and AGL(n, 2). This bound was only recently slightly exceeded. Fripertinger implemented computation of cycle indexes for GL(n, q) and AGL(n, q); if q = 2 this implementation works for about n 5≤21. By introducing appropriate precomputed tables, we reduced the cycle index computation to evaluation of a sum over partitions of n for all the four groups. Using this more efficient procedure, we obtained values of U n , V n and the explicit cycle index expressions for larger values of n.

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